1. Algebra 1B
Chapter 3 Notes

2. 3.1—Solving Equations by Adding and Subtracting
An example of an expression is x + 3.
An equation is x + 3 = 8.
What is the difference?

3. In order to solve one-step equations, we have to remember our inverses.
What is the inverse of addition? Subtraction
What is the inverse of subtraction? Addition
So in doing one-step equations, we have to remember to do the inverse and to do it to both sides.

4. Example: x + 15 = 28
No matter what inverse we do, we have to do it to both sides.
Since we are adding originally ,we need to subtract 15 from both sides.
This leaves us with x = 13.

5. Team Huddle
x + 19 = 48
18 + x = 40

6. Example: x - 15 = 28
No matter what inverse we do, we have to do it to both sides.
Since we are subtracting originally, we need to add 15 to both sides.
This leaves us with x = 43.

7. Team Huddle
x – 7 = -23
23 – x = 40

8. Sometimes there will be a word problem
Sometimes there will be a word problem. We will have to take the information presented to create an equation.
Example: Eoz is raising money to go to Florida. She has raised $80 so far. How much more money does she need to raise if she has a goal of $275.
We can set this up as an equation: x = 275.
Next, subtract 80 from both sides to get x = 195.
Therefore, Eoz needs to raise $195 to achieve her goal.

9. Team Huddle Solve the following problem in your groups.
A running back has rushed for 150 yards through three quarters. How many yards would he have to gain to match his total from the previous week, which was 224 yards?

10. 3.2—Solving Equations by Multiplying and Dividing
In order to solve one-step equations, we have to remember our inverses.
What is the inverse of multiplication? Division
What is the inverse of division? Multiplication
So in doing one-step equations, we have to remember to do the inverse and to do it to both sides.

11. Example: 15x = 120
No matter what inverse we do, we have to do it to both sides.
Since we are multiplying originally ,we need to divide 15 from both sides.
This leaves us with x = 8.

12. Team Huddle
-6y = 54
72 = 18e

13. Example:
No matter what inverse we do, we have to do it to both sides.
Since we are dividing originally ,we need to multiply both sides by 150.
This leaves us with x = 900.

14. Try on your own

15. Multiplying with Reciprocals
Remember that a reciprocal is when you flip the fraction.
For example: The reciprocal of 2/3 is 3/2.
So if given a problem like 2/3x = 18, what would we multiply both sides by to solve for x?

16. Fun with Word Problems
Egag wants to make a 300 mile trip in 6 hours. What speed should he average?
We can set this up as an equation using the formula rt=d, where r = rate, t = time, and d = distance.
Substitute 6 in for t and 300 in for distance to get 6r = 300.
What is the final step?

17. Team Huddle
Cookies are being made for you and 5 of your friends. If each person walks out with a dozen cookies, how many cookies will need to be made in all?

18. 3.3—Solving Two-Step Equations
In your team huddle, take 60 seconds and discuss what the four inverses are in terms of operations.

19. Now with one-step equations, we only had to do one operation, but now with two-step equations, we will have to do how many operations?
First we will add or subtract, then we will multiply or divide.
Example: 5x + 18 = 73

20. Team Huddle

21. Fun With Word Problems
Example: Eliza purchased 3 cases of soda and $25 worth of pizzas. If her total bill was $43, how much did she pay for a case of soda.

22. 3.4—Solving Equations with Variables on Both Sides
In your groups, huddle up and discuss what you think the term “multistep” means. How might this be different than the previous types of equations we’ve done in chapter 3?

23. Example: 8x + 16 = 4x - 12

24. Team Huddle
12x – 15 = 7x

25. In the problems we have looked at, we have had exactly one solution.
There is the possibility that we could solve a problem and have no solution or infinite number of solutions.

26. 6m – 5 = 7m + 7 – m
8(x + 3) = x

27. 8.1—Relating Decimals, Fractions, and Percents
In your groups, determine the following:
1. Would you rather eat 7/12 of a pizza, or 11/14 of a pizza?
2. Convert .271 into a fraction.

28. If Matt takes a 5 question quiz and gets four questions right, we are able to determine that he got an 80%.
A percent is a ratio that compares a number to 100.
In the case of our friend Matt, this means that if he took a 100 question quiz, he would get how many right?

29. When you get your scores back from me, it is usually a number out of another number.
Example: Matt got a 4/5.
In order to determine a percentage, we take the numerator and divide it by the denominator.
Doing this will give us a what?

30. Once we have a decimal, we have two options to convert a decimal to a percent
1. Move the decimal point two spaces to the right.
2. Multiply the decimal by 100.

31. Team Huddle
Convert 9/15 to a percent.

32. So earlier we said to convert a decimal to a percent, we move it two places to the right or multiply by 100.
To move a percent to a decimal, either move the decimal point two places to the left or divide by 100.
If there is not a decimal point shown, place it between the last number and the percent sign.

33. Example
Convert 45% to a decimal.

34. Team Huddle
Convert 121.5% to a decimal.

35. 8.2—Finding Percents
In your groups, huddle up and complete the following problems:
1. Convert 9/16 into a percent.
2. Convert 20% into a fraction.

36. If there are 8 hot dogs in a pack and you eat half of them, how many did you eat?
Well instead of saying you ate half of them, you could say you ate what percent of them?

37. If we know a percentage and we know the entire amount of something, we can solve for the partial amount.
% * of = is (This formula will be very important over the next couple of lessons)
In order to solve these problems, we first have to convert a percent into a decimal.

38. Example
50% of 8 is what number?

39. Team Huddle
28% of 300 is what number?

40. In order to figure out what percent one number is of another, we will take our % * of = is formula and get % by itself.
Thus, the formula to find a % given two numbers is

41. Example
What percentage of 80 is 56?

42. Team Huddle
What percent of 150 is 10?

43. 8.3—Finding a Number When the Percent is Known
In your groups, complete the following problems:
1. What percent of 20 is 18?
2. 70% of 560 is what number?

44. Solving for of
If we know that a number is a percentage of another number, we can solve for the second number.
To solve for of, we take

45. Example
70 is 40% of what number?

46. Team Huddle
182 is 25% of what number?

47. So now to review, there are three types of percent problems.
In the first type, we solve for the percent (we know is and of)
In the second type, we solve for of (the whole value) (we know % and is)
In the third type, we solve for is (the partial value) (we know % and of)

48. The trick is to read the problem carefully and figure out the two numbers that you know (HINT: If one of the numbers is a percent, it generally will have this symbol--%)

49. Example
25 is 40% of what number.

50. Team Huddle 1. What percent of 290 is 116?
is 40 percent of what number?
3. 40% of 290 is what number?